Problem 55 - College Algebra

Question

state the name of the property illustrated. $$ 2(-8+6)=-16+12 $$

Step-by-Step Solution

Answer

The property illustrated in the example is the 'Distributive Property'.

Step 1: Identify the operations involved

The operations involved in the expression are multiplication and addition. The expression is multiplying 2 with the sum of -8 and 6.

Step 2: Notice the reordering of terms

In the right hand side of the equation, the terms have been reordered so that each term from the sum in brackets is multiplied by 2: \(2 \times -8 = -16\) and \(2 \times 6 = 12\). Then, these multiplied values are summed up to form the equivalent equation. This kind of operation, known as distributive property, allows multiplication to 'distribute' over addition within brackets.

Step 3: Identify the property illustrated

As observed, the property applied in this mathematical expression is the distributive property of multiplication over addition. Hence, the name of the property illustrated in this example is 'Distributive Property'.

Key Concepts

Multiplication Over Addition

At its core, the concept of multiplication over addition is best understood through the distributive property, which allows us to multiply a single term by a group of terms added together within parentheses. When you encounter an expression like \( 2(-8 + 6) \), the distributive property tells us we can 'distribute' the multiplication of 2 to both -8 and 6 separately. This means performing two individual multiplications, \( 2 \times -8 \) and \( 2 \times 6 \), and then adding the results together.

By using this property, it simplifies complex expressions and makes calculations easier to handle, especially when dealing with larger numbers or variables. The distributive property is a rule that truly shows the interconnectedness of multiplication and addition in algebra.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that represent mathematical relationships. An example of a simple algebraic expression is the previous one, \( 2(-8 + 6) \). It may not seem too complex, but understanding how to handle such expressions is fundamental in algebra. Expressions can get much more complicated with multiple terms and variables involved.

In maneuvering algebraic expressions, recognizing patterns and applying properties of operations, like the distributive property, become invaluable skills. These expressions are the building blocks that lead to equations and, ultimately, to the solution of algebraic problems. Being comfortable with expressions allows deeper exploration of functions, models, and real-world mathematical applications.

Properties of Operations

Properties of operations are the rules that govern the way numbers interact with each other; they are the bedrock of all mathematical calculation and problem-solving. Key properties include commutative, associative, identity, and the distributive property we've been discussing. Each property has its own specific role, and understanding how to apply them, singly or in combination, is essential for working through algebraic expressions.

These operational properties are not just academic concepts; they're practical tools that make the work of simplifying, factoring, and expanding algebraic expressions much more efficient. Recognize that mastering these properties is not just about understanding the 'what,' but also the 'why' behind the mathematical operations. This deep understanding enables students to approach math exercises with more confidence and skill.